On the existence of -way -homogeneous Latin trades

A μ -way Latin trade of volume s is a collection of μ partial Latin squares T 1 , T 2 , … , T μ , containing exactly the same s filled cells, such that, if cell ( i , j ) is filled, it contains a different entry in each of the μ partial Latin squares, and such that row i in each of the μ partial Latin squares contains, set-wise, the same symbols, and column j likewise. It is called a μ -way k -homogeneous Latin trade if, in each row and each column, T r , for 1 ≤ r ≤ μ , contains exactly k elements, and each element appears in T r exactly k times. It is also denoted as a ( μ , k , m ) Latin trade, where m is the size of the partial Latin squares. roduce some general constructions for μ -way k -homogeneous Latin trades, and specifically show that, for all k ≤ m , 6 ≤ k ≤ 13 , and k = 15 , and for all k ≤ m , k = 4 , 5 (except for four specific values), a 3 -way k -homogeneous Latin trade of volume k m exists. We also show that there is no ( 3 , 4 , 6 ) Latin trade and there is no ( 3 , 4 , 7 ) Latin trade. Finally, we present general results on the existence of 3 -way k -homogeneous Latin trades for some modulo classes of m .